# Fixed-parameter tractability of counting small minimum $(S,T)$-cuts

1 GALaC - LRI - Graphes, Algorithmes et Combinatoire (LRI)
LRI - Laboratoire de Recherche en Informatique
Abstract : The parameterized complexity of counting minimum cuts stands as a natural question because Ball and Provan showed its #P-completeness. For any undirected graph $G=(V,E)$ and two disjoint sets of its vertices $S,T$, we design a fixed-parameter tractable algorithm which counts minimum edge $(S,T)$-cuts parameterized by their size $p$. Our algorithm operates on a transformed graph instance. This transformation, called drainage, reveals a collection of at most $n=\left| V \right|$ successive minimum $(S,T)$-cuts $Z_i$. We prove that any minimum $(S,T)$-cut $X$ contains edges of at least one cut $Z_i$. This observation, together with Menger's theorem, allows us to build the algorithm counting all minimum $(S,T)$-cuts with running time $2^{O(p^2)}n^{O(1)}$. Initially dedicated to counting minimum cuts, it can be modified to obtain an FPT sampling of minimum edge $(S,T)$-cuts.
Document type :
Reports
Domain :

https://hal.archives-ouvertes.fr/hal-02176346
Contributor : Joanna Tomasik <>
Submitted on : Monday, July 8, 2019 - 10:20:33 AM
Last modification on : Wednesday, July 10, 2019 - 1:18:36 AM

### Identifiers

• HAL Id : hal-02176346, version 1
• ARXIV : 1907.02353

### Citation

Pierre Bergé, Benjamin Mouscadet, Arpad Rimmel, Joanna Tomasik. Fixed-parameter tractability of counting small minimum $(S,T)$-cuts. [Research Report] LRI. 2019. ⟨hal-02176346⟩

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